So, take every type in the game. then create the list of types to have on the super-coverage team. 12 types max, must have at least one type supereffective against a type per type. I'm not making this clear am I? Just a list of types for a team which would have at least one Pokemon supereffective for dealing with each type. if you want to be super overachiever and do existing dual typings, then you will get upvoted, but not entirely necessary, just awesome. There are probably multiple answers, so you can include multiple.

Quick word of advice, if the intent is to create a team out of Pokemon with these types, it's probably not worth it. It doesn't consider that a. you can have defensive weaknesses as well (especially combinations of types) and b. typing is far from the only factor that determines how well a team covers the metagame. It also restricts your team building to Pokemon of certain types which sounds more to me like a burden.
But feel free to ignore me if this isn't what you mean at all.

2 Answers

To start answering this, there are three types that are absolutely required on any list for this question.

Ground

Since this is the only type that is super-effective against Electric types.
It also hits Fire, Poison, and Steel types.

Fighting

This is the only type that is super-effective against Normal types.
It also hits Dark, Rock, Steel, and Ice types.

Dark or Ghost

These types are super-effective against the same types, although their resistances are slightly different. They both deal with Psychic and Ghost types.

This means the remainder of the team would need to be composed of types to deal with Grass, Water, Fighting, Ground, Flying, Bug, Dragon, and Fairy types.

Ice

I think this is the next logical type to add, as it deals with quite of few of the leftover types. This means you can cross Grass, Ground, Flying, and Dragon types out of the above list.

At this point, the only types you have left to deal with is Fighting, Bug, Fairy, and Water. These types only have one shared weakness.

Flying

This is super-effective against Bug and Fighting types.

Water and Fairy don't have any shared weaknesses, so you will need to add two more types, one to cover each of them.

Electric or Grass

These both cover Water types.

Poison or Steel

These both cover Fairy types.

Summary

I know you said a maximum of 12 types, but 7 is enough to cover everything. These types are:

Ground

Fighting

Dark/Ghost

Ice

Flying

Electric/Grass

Poison/Steel

The first four I feel are required for any team needed to cover all types. The last 3 can no doubt be swapped around, though. I really don't want to even think about adding dual-types into the mix, so I will leave my answer here.

These are the smallest ones without freeze-dry in case you don't think freeze-dry counts as a type. Note that all of them have grass because there's no freeze-dry to cover water ground. Also note that none of them are pieces of sheet music.